A206943 Generalized repeat unit one numbers: all numbers of the form (m^p-1)/(m-1), where abs(m) > 1 and p is odd prime.

3, 7, 11, 13, 21, 31, 43, 57, 61, 73, 91, 111, 121, 127, 133, 157, 183, 205, 211, 241, 273, 307, 341, 343, 381, 421, 463, 507, 521, 547, 553, 601, 651, 683, 703, 757, 781, 813, 871, 931, 993, 1057, 1093, 1111, 1123, 1191, 1261, 1333, 1407, 1483, 1555, 1561

Offset: 1

Author

Lei Zhou, Feb 28 2012

Keywords

• nonn
• base
• easy

Comments

Here we define "generalized repeat unit one numbers" as numbers that can be represented in the form 11...1_m where the number of ones is k > 2 and |m| > 1.

Normal repeat unit one numbers (a.k.a. "repunits") are numbers in the form 11...1_10 with 2 or more ones.

Trivially, any number n = 11_(n-1).

These numbers take the form of cyclotomic polynomial number Phi(k,m) with k in the form 2^i*p^j, where p is prime and i >= 0, j >= 1. It has p digits of one base -m^(2^(i-1)*p^(j-1)) when i > 0 or base m^(p^(j-1)) when i = 0.

This sequence is a subsequence of A206942.

Examples

			111_(-2) = 3, so 3 is a term;
111_2 = 7, so 7 is a term;
11111_(-2) = 11, so 11 is a term.
3 = (2^3 + 1)/(2 + 1);
7 = (2^3 - 1)/(2 - 1) = (3^3 + 1)/(3 + 1);
11 = (2^5 + 1)/(2 + 1).
		

Crossrefs

Cf. A206942, A208507.

Programs

Mathematica

phiinv[n_, pl_] := Module[{i, p, e, pe, val}, If[pl == {}, Return[If[n == 1, {1}, {}]]]; val = {}; p = Last[pl]; For[e = 0; pe = 1, e == 0 || Mod[n, (p - 1) pe/p] == 0, e++; pe *= p, val = Join[val, pe*phiinv[If[e == 0, n, n*p/pe/(p - 1)], Drop[pl, -1]]]]; Sort[val]]; phiinv[n_] := phiinv[n, Select[1 + Divisors[n], PrimeQ]]; maxdata = 1600; max = Ceiling[(1 + Sqrt[1 + 4*(maxdata - 1)])/4]*2; eb = 2*Floor[(Log[2, maxdata])/2 + 0.5]; While[eg = phiinv[eb]; lu = Length[eg]; lu == 0, eb = eb + 2]; t = Select[Range[eg[[Length[eg]]]], ((EulerPhi[#] <= eb) && (a = FactorInteger[#]; b = Length[a]; (((b == 1) && (a[[1]][[1]] > 2)) || ((b == 2) && (a[[1]][[1]] == 2))))) &]; ap = SortBy[t, Cyclotomic[#, 2] &]; an = SortBy[t, Cyclotomic[#, -2] &]; a = {}; Do[i = 0; While[i++; cc = Cyclotomic[ap[[i]], m]; cc <= maxdata, a = Append[a, cc]]; i = 0;  While[i++; cc = Cyclotomic[an[[i]], -m]; cc <= maxdata, a = Append[a, cc]], {m, 2, max}]; Union[a]
nn = 40; ps = Prime[Range[2, PrimePi[Log[2, 2*nn^2 + 1]]]]; t = {}; Do[If[Abs[m] > 1, n = (m^p - 1)/(m - 1); If[n < nn^2, AppendTo[t, n]]], {p, ps}, {m, -nn, nn}]; t = Union[t] (* T. D. Noe, May 03 2013 *)

Extensions

Name improved and new example added by Thomas Ordowski, May 03 2013